a coupled 1d-2d hydrodynamic model for urban flood inundation

Research ArticleA Coupled 1D-2D Hydrodynamic Model forUrban Flood Inundation

Yuyan Fan,1 Tianqi Ao,1 Haijun Yu,2 Guoru Huang,3 and Xiaodong Li1

1College of Water Resources and Hydropower, State Key Laboratory of Hydraulics and Mountain River Engineering,Sichuan University, Chengdu 610065, China2State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin,Research Center on Flood and Drought Disaster Reduction of the Ministry of Water Resources,China Institute of Water Resources and Hydropower Research, Beijing 100038, China3State Key Laboratory of Subtropical Building Science, School of Civil Engineering and Transportation,South China University of Technology, Guangzhou 510640, China

Correspondence should be addressed to Haijun Yu; [email protected]

Received 19 January 2017; Revised 22 April 2017; Accepted 30 April 2017; Published 23 May 2017

Academic Editor: Hongkai Gao

Copyright © 2017 Yuyan Fan et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Hydrodynamic models were commonly used for flood risk management in urban area. This paper presents initial efforts indeveloping an urban flood inundation model by coupling a one-dimensional (1D) model with a two-dimensional (2D) modelto overcome the drawbacks of each individual modelling approach, and an additional module is used to simulate the rainfall-runoff process in study areas. For the 1D model, the finite difference method is used to discretize the Saint-Venant equations. Animplicit dual time-stepping method (DTS) is then applied to a 2D finite volume model for an inundation simulation to improvecomputational efficiency. A total of four test cases are applied to validate the proposed model; its performance is demonstrated by acomparison with an explicit scheme and previously published results (an extensive physical experiment benchmark case, a verticallinking example, and two real drainage cases with actual topography). Results demonstrate that the proposedmodel is accurate andefficient in simulating urban floods for practical applications.

1. Introduction

Over the past decades, there have been an increasing numberof extreme rainfall andflood events occurring globally in rela-tion to climate change, which paralyze cities and result in seri-ous social and economic consequences. In addition, small-scale urban flooding occurs frequently throughout the worldin relation to poor drainage. Consequently, there is a growingneed for storm water management strategies to reduce urbanflood risk.

Hydrodynamic models are solid tools in urban stormwater management. Numerical simulations of urban floodsare important when scientifically planning and designingurban drainage systems and providing efficient urban flooddisaster control and management strategies. Although manymodels have been developed for river and coastal flooding[1–3], urban flooding models have not yet been adequately

developed; this partly attributes to the complex flowprocessesin urban areas when inundation occurs. As there is a one-dimensional (1D) flow property in sewer systems, the highlyefficient one-dimensional (1D) model is the most commonlyused tool to simulate the hydraulic performance of urbandrainage systems. A number of sewermodels are used widely,such asMOUSE [4], InfoworksCS [5], and StormWaterMan-agement Model (SWMM) [6]. However, 1D models can onlypredict the surcharge volume froma drainage system in termsof the ground’s surface. Therefore, to accurately simulatedetailed urban flood propagation and inundation of theground’s surface, coupling of the 1D and 2D hydrodynamicmodel is necessary to consider flow interactions betweenunderground pipes and the ground’s surface, and the 1Dmodel and 2Dmodel in the coupled model are used to simu-late flows in the pipe/river drainage system and surface inun-dation, respectively. In recent years, several coupled 1D-2D

HindawiAdvances in MeteorologyVolume 2017, Article ID 2819308, 12 pageshttps://doi.org/10.1155/2017/2819308

https://doi.org/10.1155/2017/2819308

2 Advances in Meteorology

models for urban floods have been developed and applied [7–10].

2D models are widely used to simulate urban flood inun-dation [11–14]. However, 2D calculation can be time consum-ing, and large variations in cell sizes may occur because of thecomplex urban underlying surface or local refinement, whichleads to lower efficiency because the time step is limited bynumerical stability and is largely determined by the smallestcells in the mesh. Although the time steps of 1D and 2Dmodels are both limited by numerical stability, the time stepsadopted in 1D models can usually be much larger than thoseof 2Dmodels. However, most existing 1D-2D coupledmodelsrequire the same time step in 1D and 2D models, which mayresult in lower efficiency of the 1D models. To improve thedrawback of the inefficient 2D model, large-sized grids andsimplified models are commonly adopted, which can thenlead to a loss of accuracy.

The aim of this paper is to develop a coupled modelfor urban flood inundation by using an implicit dual time-stepping method (DTS) [15]. DTS is adopted to a 2D modelto improve run time efficiency. The method is fully implicit,matrix-free, and easily implemented, meanwhile providingthe possibility of parallel computing. With the application ofthe DTS, arbitrarily large time steps of the 2D model are per-mitted and computational efficiency of both 1D and 2Dmodelis significantly improved. In addition, the coupled model hasgreater stability due to the use of an implicit scheme andis therefore particularly suitable for urban flood simulationswhere there is a complex underlying surface.

In this study, the 1D and 2D governing equations for freesurface and pressurised flows are presented together withdetails of numerical schemes. The proposed model is thenvalidated against four numerical test problems, and in thefinal section the study is summarized and concluded.

2. Numerical Schemes

2.1. Rainfall-Runoff Model. To simulate the rainfall-runoffprocess in urban areas, the study area is firstly divided intosubcatchment areas; these represent land areas that collectrainwater and allow infiltration and drainage to a specificnode.These subcatchments are then divided into imperviousand pervious areas to enable the separate calculation ofrunoff, and the Horton infiltration method [16] is used tosimulate the infiltration process in pervious area. Rainwatercollected by the subcatchments finally drains to the drainagenetwork, and the 1D model is used to calculate the flowrouting in the drainage network.

2.2. 1D Model. The 1D model is used to simulate flows inthe urban drainage systemwhere free surface and pressurisedflows coexist. Free surface flow can be expressed mathemati-cally by the 1D shallow-water equation (SWE), and if a ficti-tious Preissmann slot is introduced, the same set of governingequations can be used for full pipe flowwithout a free surface,

known as pressurised flows. The continuity and momentumequations of the 1D model are given by

𝜕𝐴𝜕𝑡 + 𝜕𝑄𝜕𝑥 = 0,𝜕𝑄𝜕𝑡 + 𝜕𝜕𝑥 (𝛽𝑄

2

𝐴 ) + 𝑔𝐴𝜕𝑍𝜕𝑥 + 𝑔𝐴𝑆𝑓 = 0,(1)

where 𝑡 denotes time; 𝑥 is the Cartesian coordinates; 𝑄 isdischarge; 𝐴 is the wet cross-section area; 𝑍 is the water leveland water head for free surface flow and pressurised flow,respectively;𝑔 is gravity accelerationwith a value of 9.81m/s2;𝛽 is the momentum correction factor; and 𝑆𝑓 is the frictionslope.

Firstly, the governing equations are solved by the finitedifference method using Preissmann’s four-point implicitdifference scheme in individual channels or pipes. The dis-cretization of Preissmann’s scheme is as follows:

𝜕𝑓𝜕𝑡 = Φ(𝑓𝑛+1𝑠+1 − 𝑓𝑛𝑠+1Δ𝑡 ) + (1 − Φ)(𝑓𝑛+1𝑠 − 𝑓𝑛𝑠Δ𝑡 ) ,

𝜕𝑓𝜕𝑥 = 𝜃(𝑓𝑛+1𝑠+1 − 𝑓𝑛+1𝑠Δ𝑥 ) + (1 − 𝜃) (𝑓𝑛𝑠+1 − 𝑓𝑛𝑠Δ𝑥 ) ,

𝑓 (𝑥, 𝑡) = 𝜃2 (𝑓𝑛+1𝑠+1 + 𝑓𝑛+1𝑠 ) + (1 − 𝜃)2 (𝑓𝑛𝑠+1 + 𝑓𝑛𝑠 ) ,

(2)

where Φ and 𝜃 are weighting coefficients, Δ𝑡 and Δ𝑥 are thetime step and space step, respectively, 𝑛 is the index of timelevel, 𝑠 is the index of locations, and𝑓 is the general function.

Then, the water level and flow discharge at each sectionof a channel or pipe can be denoted by using water level atthe junctions (as shown in Figure 1) if a recursive methodis adopted. Flow discharge and the water level at junctionscan be solved by adopting an iteration procedure. Finally, thewater level and flow discharge of each section are updated byapplying the obtained state value of junctions.

However, Preissmann’s scheme is not able to simulatesupercritical flows, and to overcome this limitation the super-critical flow is treated as a subcritical flow by reducing theinfluence of the convective momentum term in the momen-tum equation. The convective momentum term can beexpanded to two parts,

𝜕𝜕𝑥 (𝑄2

𝐴 ) = 𝑄𝜕𝑢𝜕𝑥 + 𝑢𝜕𝑄𝜕𝑥 , (3)

where 𝑢 is the speed of a section.A reduction coefficient based on the Froude number is

then introduced to gradually reduce the first part of theconvective momentum term.

2.3. 2D Model. The 2D model for urban surface flood inun-dation simulation is based on two-dimensional shallow-waterequations in a conservative form. If we neglect wind effectsand the Coriolis term, the governing equations can be writtenas 𝜕U𝜕𝑡 + 𝜕E𝜕𝑥 + 𝜕G𝜕𝑦 = S, (4)

Advances in Meteorology 3

Junction

Pipe or channel

Section

Junction E

Junction S

Section J

Qj = f(ZS, ZE)

Zj = f(ZS, ZE)

Figure 1: Schematic illustration of pipe network and flow variables.

where 𝑡 denotes time; 𝑥 and 𝑦 are the Cartesian coordinates;U, E, and G denote vectors of the conserved flow variablesand fluxes in the𝑥- and𝑦-directions, respectively; and S is thesource vector containing the bed slope source S𝑏 and frictionsource S𝑓. These vectors are expressed as

U = [[[ℎℎ𝑢ℎV]]],

E = [[[[[

ℎ𝑢ℎ𝑢2 + 𝑔ℎ22ℎ𝑢V

]]]]],

G = [[[[[

ℎVℎ𝑢V

ℎV2 + 𝑔ℎ22

]]]]],

S = S𝑏 + S𝑓 = [[[[

0𝑔ℎ (𝑆𝑜𝑥 − 𝑆𝑓𝑥)𝑔ℎ (𝑆𝑜𝑦 − 𝑆𝑓𝑦)

]]]],

(5)

where ℎ, 𝑢, V, and 𝑏 are the water depth, depth-averagedvelocity in the 𝑥- and 𝑦-directions, and bottom elevation,respectively; 𝑆𝑜𝑥 and 𝑆𝑜𝑦 are bed slopes in the 𝑥- and 𝑦-directions, respectively; and 𝑆𝑓𝑥 and 𝑆𝑓𝑦 are friction slopesin the 𝑥- and 𝑦-directions, respectively.

A Godunov-type finite volume method with an HLLCflux function is adopted and the variables are located at thegeometric centres of cells. The 2D SWE is integrated over thecontrol volume and expressed as

∫Ω

𝜕U𝜕𝑡 𝑑Ω + ∫Ω(𝜕E𝜕𝑥 + 𝜕G𝜕𝑦 )𝑑Ω = ∫

ΩS𝑑Ω. (6)

Using the second-order backward difference scheme ateach cell in time discretization and applying Green’s theorem,(6) can be written as

3U𝑛+1𝑖 − 4U𝑛𝑖 + U𝑛−1𝑖2Δ𝑡 + 1𝐴 𝑖 (3∑𝑘=1

F𝑖,𝑘 ⋅ n𝑖,𝑘𝑙𝑖,𝑘 − 𝐴 𝑖S𝑖)= 0,

(7)

where 𝐴 𝑖 is the area of cell i; n is the unit outward vectornormal to the boundary; 𝑘 and 𝑙 are the index and lengthof the faces of cell i, respectively; F𝑖,𝑘 is the numerical fluxthrough the edge; 𝑛 is the physical time level; S𝑖 is the sourceterm of cell i; U𝑛𝑖 and U𝑛−1𝑖 are the solutions from the currentand previous time steps; andU𝑛+1𝑖 is the unknown solution ofthe next time step to be solved.

Equation (7) can be solved by introducing a pseudo-timevariable 𝜏,

𝑑U𝑖𝑑𝜏 + RES (U𝑖) = 0, (8)

RES (U𝑖) = 3U𝑛+1𝑖 − 4U𝑛𝑖 + U𝑛−1𝑖2Δ𝑡+ 1𝐴 𝑖 (

3∑𝑘=1

F𝑖,𝑘 ⋅ n𝑖,𝑘𝑙𝑖,𝑘 − 𝐴 𝑖S𝑖) .(9)

The process of iteratively solving (8) is known as inneriteration. As the inner steady problem converges, the residualterm RES(U𝑖) converges to zero and the solution of (7)advances from U𝑛𝑖 to U𝑖

𝑛+1. The inner steady problem can besolved using a variety of schemes, and an effective implicitmatrix-free LU-SGS algorithm [17] is adopted in this study.To obtain high efficiency and prevent possible endless loops, aconvergence tolerance and a maximum number of iterationsare used to control the inner iteration scheme. The iterativeprocess is terminated only if the residual term is less than theconvergence tolerance or the number of iterations exceeds themaximum number of iterations.


GTD: water flows from ground DTG: water flows from sewer drainagesystem to ground surface

Ground surface

surface into sewer drainage system

Manhole

Drainage pipe

Figure 2: Schematic illustration of flow exchange between surface and subsurface systems.

3. Interaction between the Sewer andOverland Flow

In this study, the manhole (junction) function points of theflow exchange between the surface and subsurface systems(as shown in Figure 2). Surcharge occurs when water flowsfrom the sewer drainage system to the ground surface system(DTG) at a time when the water level in the manhole reachesground elevation. In addition, water flows from the ground’ssurface into the sewer drainage systems (GTD) when thewater level, or the water head in a manhole, is lower than theground water level.

There is currently no specific theory or widely acceptedmethods used to accurately describe the overflow frommanhole to surface. Therefore, in this study the discharge ofDTG is computed during the iteration process of solving themanhole water levels and is given by

𝑄𝑛𝑠 = 𝑚∑𝑟=1

𝑄𝑟 − 𝐴nod (𝑍𝑛+1nod − 𝑍𝑛nod)Δ𝑡 , (10)

where 𝐴nod is the area of the manhole; 𝑍𝑛+1nod and 𝑍𝑛nod arethe water level at the current and previous time steps, respec-tively; 𝑄𝑟 is the inflow or outflow of the 𝑟th branch aroundthe manhole; and 𝑚 is the number of the branches aroundthe manhole.

The discharge of GTD through manholes can bedescribed using either a weir equation or an orifice equation[3, 18]. In this paper, the weir equation, which handles bothfree and submerged flows, is adopted and is given by𝑄𝑠𝑛

= {{{{{{{𝑚𝐶nodℎsur√2𝑔ℎsur if

ℎnodℎsur ≤23

𝑚𝐶nodℎnod√2𝑔 (ℎsur − ℎnod) if 23 < ℎnodℎsur ≤ 1,(11)

where𝑚 is the discharge coefficient; 𝐶nod is the perimeter ofthe node; ℎsur and ℎnod are the water depth in the node andground surface above the top of the node, respectively; and𝑄𝑠𝑛 is the exchange discharge.

Elevation (m)

7.5 7.7 7.9 8.1 8.3 8.5

Upstream

Downstream

Figure 3: Schematic diagrams of Toce River experiment examples.

4. Application and Discussion

In this section, the capability of the proposed model is vali-dated using four test case applications: one extensive physicalexperiment benchmark case, one vertical linking exampleand two real drainage cases with actual topography.

4.1. Toce River Experiment with Blocks for 2DModel. In urbanareas, densely spaced buildings have a significant effect onflood propagation and inundation. To provide validation datafor urban flood models, a series of experiments were con-ducted for the IMPACT (Investigation of extreMe flood Pro-cesses And unCertainTy) project by CESI (Centro Elettrotec-nico Sperimentale Italiano) [19]. These experiments wereextensively applied to test the applicability of models in highdensity urban areas [11, 20, 21]. In the experiments, a clusterof blocks was placed in an instrumented physical model of ashort reach of the Toce River valley in Italy on a 1 : 100 scaleto simulate the blockage effects of an urban district.

One set of experiments was used to verify the 2D modelpresented in this paper, and the terrain of the domain andthe distribution of blocks are illustrated in Figure 3. Althoughthe possibility of a very thin film of water or moisture wettingthe concrete bed due to previous runs during the experimentexists, the water film has little effect on the results, so thewhole domain was assumed initially dry and in the giveninflow boundary (illustrated in Figure 4) the model was run


10 20 30 40 50 600t (s)

0

20

40

60

80

100

Q(L

/s)

Figure 4: Inflow boundary conditions for upstream.

for 60 s. Experimental data from four observation stationswere used for validation. Two stations (P3 and P4) locatedat the upper reaches of the blocks and two stations (P9 andP10) located between the building and the downstream of theblocks, respectively. Figure 5 shows the model’s predictionsand experimental measurements at these four observationstations. The numerical results of the water depth at theselocations are in good agreement with the experimental dataand the predictions of a Godunov-type 2D flood modeldeveloped by Kim et al. [21].

The explicit MUSCL-Hancock scheme (MHS) is moreefficient than the commonly used Runge-Kutta scheme andis thus chosen here to be representative of explicit schemes.The essential idea of the MUSCL-Hancock scheme is to usecell averages to predict the values of the conserved quantitiesat cell edges at 𝑡 + Δ𝑡/2 and then use these predicted valuesto update the solution to 𝑡 + Δ𝑡 [22, 23]. A comparison wasmade between theDTS and theMHS, where the adaptive stepmethod was used to obtain the most efficient performance.Courant number was used to determine the time stepsdynamically for both MHS and DTS. Table 1 shows therelative run time of various cases computed byDTS andMHSwith different Courant numbers, and a comparison of numer-ical results is presented with different Courant number inFigure 6. It can be seen from Figure 6 and Table 1 that the runtime of DTS are only 37% of that determined with MHS withno other significant differences observed. Courant numbermust be less than 1.0 for an explicit scheme and thus thetime steps of the MHS are largely limited. Because DTS isunconditionally stable, large time steps can be adopted. Theresults also suggested that the efficiency of DTS improved asthe Courant number increased; however, the effect was notobvious over a certain range.

4.2. Real Drainage Case for 1D Model. To verify the ability,reliability, and precision of the 1D model presented in thispaper in relation to dealing with an actual urban drainageproblem, a real drainage system of a small community inChangsha City, China, was simulated. Results of the presentmodel were then compared with those of the SWMMmodel[6], which is one of the most complete and widely used

Table 1: Relative run time of various cases.

Scheme Courant number Relative run timeMSH 0.8 1.00DTS 1.0 1.32DTS 2.0 0.74DTS 3.0 0.48DTS 4.0 0.39DTS 5.0 0.37

models globally for planning, analysis, and design related todrainage systems within urban areas. The community wasbuilt in the 1990s and covers a total area of 11.7 hectares,56% of which are impermeable, and the average gradient is0.3%. As illustrated in Figure 7, the entire area was dividedinto 23 subcatchments, and 23 trunk pipes were identified formodelling.

Two designed storms (labelled R1 and R2) were calcu-lated by the model, and Figure 8 illustrated the dischargehydrographs of the outfall.The total rainfall of R1 and R2 was58mm and 75mm, respectively. It can be seen from Figure 8that the results of present 1D model in this paper are almostidentical with the results of SWMM model, which indicatedthat the proposed model has reliable numerical accuracy. Itshould be noted that two models have some nuances in thecalculation of the flow peak, which can possibly be attributedto the different numerical methods used in the two models.

4.3. Vertical Linking Case. A vertical linking case wasdesigned to assess the rationality and feasibility of thecoupling strategies of the 1D and 2Dmodels presented in thispaper. As shown in Figure 9, the pipe network was formed bysix pipes and six nodes, and all nodes except Nodes 1 and 6could exchange water with the surface. The pipes and nodeswere numbered (their attributes are listed in Tables 2 and 3,resp.). A square closed floodplain with a length of 200m wassituated at the top of Nodes 2, 3, 4, and 5; the elevation of thefloodplain was 0 m. The Manning coefficient for the flood-plain was set to 0.025; the floodplain and pipe network wereinitially dry and the inlet flow at Node 1 was increased from 0to 1m3/s in the first 10 minutes and then maintained at aconstant rate, while a free boundary was given at Node 6.

The coupled model presented in this paper was used tosimulate this case, while Infoworks ICM,which is widely usedcommercial drainage software that includes a 1Dmodel and a2D model, was also used to simulate this case for a compari-son.

Figure 10 illustrates the distribution of the flow field andwater level throughout the floodplain; Tables 4 and 5 show thedischarge in each pipe and the water depth at each node aftera steady state is reached, respectively. From these figures andtables it can be seen that the water flows from the pipe to thesurface through Node 2, as the pipe diameters of Pipes 2 and3 are relatively small and the drainage capacity is insufficient,but that water flows from the floodplain to the pipe networkthrough Nodes 3, 4, and 5. The given inlet flow at Node 1maintained at a constant rate at last, so once a balance of the


MeasurementModel of Kim et al. (2014)Present model

0

2

4

6

8

10

12ℎ

(cm

)

10 20 30 40 50 600t (s)

(a)


0

2

4

6

8

10

12

ℎ(c

m)

10 20 30 40 50 600t (s)

(b)


0

2

4

6

8

10

12

ℎ(c

m)

10 20 30 40 50 600t (s)

(c)


0

2

4

6

8

10

12

ℎ(c

m)

10 20 30 40 50 600t (s)

(d)

Figure 5: Comparison of numerical results with physical experiment: (a) P3; (b) P4; (c) P9; (d) P10.

Measurement

10 20 30 40 50 600t (s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

ℎ(c

m)

DTS,DTS,

DTS,

DTS,DTS,

C = 5 C = 4C = 2C = 3

C = 1 MHS, C = 0.8

Figure 6: Comparison of numerical results with different Courant number in the station P4.


Manhole

N

OutfallPipeSubcatchment

Figure 7: Subcatchments and conduits of study area.

PrecipitationSWMMPresent model

0

1

2

3

4

30 60 90 120 150 1800t (min)

40

30

20

10

0P

(mm

)

Q(m

3/s

)

(a)

PrecipitationSWMMPresent model

0

2

4

6

8

30 60 90 1200t (min)

30

20

10

0

P(m

m)

Q(m

3/s

)

(b)

Figure 8: History of discharge change process at downstream outfall: (a) R1; (b) R2.

two kinds of flow is reached, which means the discharge ofwater flowing in and out is equal, the flow in the floodplainwas eventually stabilized. From a comparison of the resultsin these tables, it is evident that the pipe discharge and nodewater depth simulated by the two models are basically inagreement, which indicates that the coupledmodel presentedin this paper gives reliable results. The differences in theresults simulated by the models may be due to different cou-pling algorithms and meshing. However, it is considered thatoverall the model results have certain rationality.

With respect to the water volume balance, the accumu-lated inflow volume should be equal to the sum of the accu-mulated outflow volume and the water volume flowing ontothe floodplain, if the change of water volume inside the chan-nel is ignored. Similarly, the volume in the floodplain shouldbe equal to the net exchange water through four nodes.Figure 11(a) shows the history of the change process in totalvolume of GTD, total volume of DTG, and the water volume

within the floodplain. Figure 11(b) shows the history of thechange process in total inflow volume, the sum volume of theoutflow, and the water in the floodplain. From Figures 10(a)and 10(b), it can be seen that the sum of the GTD and watervolume within the floodplain is equal to the total volume ofthe DTG and that the inflow volume is equal to the sum of theoutflow volume and the water volume within the floodplain;this indicates good water conservation properties.

4.4. Real Drainage Case for Coupled 1D and 2D Model. Thecoupled 1D and 2D model proposed and established in thispaper was further tested by application to a real urbanrainstorm simulation case in the Xinhepu Community,Guangzhou City. Complete data of the terrain and pipe net-work are available for this community and it was thus selectedas a benchmark case to assist the verification of the proposedmodel. The total area of the study area is 12.27 hectares and85% of the area is impervious. Subcatchment discretization


Table 2: Pipe characteristics of network.

Pipe Upstream node Downstream node Diameter (m) Length (m) Manning’s coefficient1 1 2 1.0 100 0.0132 2 4 0.6 100 0.0133 2 3 0.6 100 0.0134 4 5 0.7 100 0.0135 3 5 0.7 100 0.0136 5 6 1.0 100 0.013

Table 3: Node characteristics of network.

Node 𝑥 𝑦 Bottomelevation

(m)

Top elevation(m)

Top diameter(m)

Bottom area(m2)

1 −70.7107 100.0000 −1.1586 0.0 1.0 4.02 29.2893 100.0000 −1.2586 0.0 1.0 0.83 100.0000 170.7107 −1.3293 0.0 1.0 0.84 100.0000 29.2893 −1.3293 0.0 1.0 0.85 170.7107 100.0000 −1.4000 0.0 1.0 0.86 270.7107 100.0000 −1.5000 0.0 1.0 0.8

Table 4: Discharge of pipes after results stabilized.

Pipe Present model(m3/s)

Infoworks ICM(m3/s)

Absolute difference(m3/s)

Relativedifference

(%)1 1.000 1.000 0.000 0.002 0.412 0.413 0.002 0.243 0.412 0.413 0.002 0.244 0.471 0.471 0.000 0.005 0.470 0.472 0.002 0.426 1.000 1.000 0.000 0.00

1 2

3

6

NodePipe

InflowOutflow

Pipe 6

Pipe 2

Pipe 15

4

Pipe 3 Pipe 5

Pipe 4

yx

(200, 200)

(200, 0)

(0, 200)

(0, 0)

Figure 9: Schematic diagrams of vertical linking case.

was based on the layout of streets, buildings, and terrainfeatures and the entire study area was divided into 94 sub-catchments. In all, 381 conduits were identified for modelling

through streamlining and consolidation; these are shown inFigure 12(a). To make an accurate calculation of the flooddepth caused by a rainstorm, the surface 2D model consid-ered only the street area; the housing area was assumed tobe watertight and thus the water flowed only onto the street.The assumption is reasonable because the floodwater will notflow into the building when the water depth is not very deep.Figure 12(b) shows the distribution and terrain of the streetsin the calculation range and it can be seen that the elevation ofthe whole area became gradually lower from north to south.

Four rainstorms with different return periods were sim-ulated to assess the drainage capacity and waterlogging risksof the community. A return period is a statistical measure-ment typically based on historic data denoting the averagerecurrence interval of the rainfall or flood over an extendedperiod of time. Details of pipes reached full drainage capacity(FDC) and surface flooding are presented in Tables 6 and7, respectively, where the following is evident. (1) With anincrease in the return period of the rainstorm, the rate of theFDC pipes improved and there was an increase in the maxi-mum velocity and flooded area of the streets. However, there


050

100150

200 050

100150

2000.1180.1200.1220.124

z(m

)

x (m) y (m)

(a)

0 50 100 150 2000

50

100

150

200

x (m)

y(m

)(b)

Figure 10: Distribution of flow field and water level in the floodplain: (a) water level; (b) flow field.

Total volume of GTDGTD Total volume of DTG

Water in the floodplain+

0

1

2

3

V(104

m3)

12 24 36 480t (h)

water in the floodplain

(a)Accumulated inflow volume

Outflow volume water in the floodplain+

0

6

12

18

V(104

m3)

12 24 36 480t (h)

(b)

Figure 11: History of water volume change process.

Table 5: Depth of nodes after results stabilized.

Node Present model(m)

Infoworks ICM(m)

Absolutedifference

(m)

Relativedifference

(%)1 1.463 1.500 −0.037 −2.542 1.388 1.396 −0.008 −0.543 1.008 0.991 0.017 1.754 1.008 0.990 0.018 1.825 0.821 0.779 0.042 5.396 0.573 0.574 −0.001 −0.17


ManholePipeSubcatchment

(a)

Elevation (m)

7 8 9 10 11 12 13 14 15 16(b)

Figure 12: Schematic illustration and computational domain: (a) subcatchments and conduits; (b) streets.

was little change in the length of time that pipes remained inFDC state; this could be because that the drainage capacity ofpartial pipeswas poor and that the FDC state occurred duringlight rainfall, thereby demonstrating that increases in rainfallintensity have little effect on the length of time that pipesremain in a FDC state. (2)When rainswith return periods of 1and 2 years occurred, a smaller number of pipes reached FDCand the water ran at a low velocity in the streets with a depthbelow 0.10 m. However, with a return period of 10 years,almost one-third pipes were at a FDC state and there wasgreater surface flooding in the streets, thereby indicating anincreased risk of flooding. However, overall, rainstorm runoffwas eliminated quickly and did not cause a large area of flood-ing, which indicates the community has good drainability.

Figure 13 shows themaximumwater depth and FDC timeof pipes of a rainstormwith a return period of 10 years. As canbe seen from Figure 13, the FDC time of pipes is mainly con-centratedwithin 15min and thewater depthwithin the streetsiswithin 0.2m,which is completely consistentwith the resultsof Table 6. The drainage network and facilities of XinhepuCommunity have been upgraded in recent years and theoverall drainage drainability is good; therefore, the results ofthe model are in good agreement with the actual situation ofthe region.

5. Discussion and Conclusion

In this paper, a coupled 1D and 2D model for urban floodinundation using an implicit DTSmethod was presented andanalysed. The following conclusions are made.

An implicit dual time-stepping method was applied tothe 2D model to improve run-time efficiency. Results showthat the run time can be reduced by more than 60% with no

significant loss of accuracy, compared to the explicit scheme.The effectiveness of the present model in simulating floodsin a dense urban area was verified through application toan experimental case. The model shows good performance,which indicates its capability of accurately and efficientlyhandling complex flows involving urban buildings.

A vertical linking case was designed to assess the rational-ity and feasibility of coupling strategies and the water volumebalance. The results computed by the present model werecompared to those obtained by Infoworks ICM; only subtledifferences were noted, which indicates that the couplingstrategies of the present model have certain rationality. Awater quantity analysis also shows that the model has goodwater conservation properties.

The effectiveness and robustness of the present modelwere verified through application to an experimental case ofan urban flood, a real drainage case for the 1Dmodel, a verti-cal linking case, and an actual case with irregular topographyand complex pipe network. The model once again showeda good performance, which indicates that it is capable ofaccurately and efficiently simulating complex water flows inurban areas.

Weir formula and orifice formula are the most commonmethods to calculate flow interaction including GTD andDTG, but they do not always give accurate results in somespecific situations, which is a problem faced by almost allurban flood models. The presented 1D-2D coupled modeladopted a weir formula to compute the discharge of GTD anda method to compute the discharge of DTG during the itera-tion process. This combination of the two methods is slightlymore stable, but there is no evidence that this combinationis more applicable than weir formula and orifice formula.Therefore, more studies about flow interaction mechanism


0

Maximum water depth (m)

0.1 0.2 0.3(a)

FDC time (min)0–55–1010–15

15–3030–70

(b)

Figure 13: Results of rainstorm with a return period of 10 years: (a) maximum water depth; (b) FDC time.

Table 6: Details of FDC pipes.

Return period(year)

Total rainfall(mm)

Maximumrainfall intensity

(mm/h)

Number of FDCpipes

Proportion ofFDC pipe (%)

Maximum timeof FDC (min)

1 53.0 157.7 45 11.81 66.52 60.0 178.5 55 14.44 69.55 69.2 206.0 100 26.25 70.010 76.2 226.8 106 27.82 71.0

Table 7: Details of ground surface flooding.

Return period (year) Flooding area (m2) Proportion of flooding street (%) Maximum street speed (m/s)ℎmax > 0.01m ℎmax > 0.10m ℎmax > 0.01m ℎmax > 0.10m1 398.40 8.73 1.68 0.04 1.172 576.70 12.58 2.43 0.05 1.335 688.36 43.37 2.90 0.18 1.5210 1291.36 126.60 5.44 0.53 1.76

and calculation method are very necessary and should beconsidered in future work.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (Project no. 50979062), the Interna-tional S&T Cooperation Projects, the Ministry of Science

and Technology of China (no. 2012DFG21780), and IWHRResearch&Development Support Program (JZ0145B222017).

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